Ground states and high energy solutions of the planar Schrödinger–Poisson system.

In this paper, we are concerned with the Schrödinger–Poisson system {−Δu+u+ϕu=|u|p−2uin Rd,Δϕ=u2in Rd. Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case . In contrast, much less information is available in the planar case which is the focus of the present paper. It has been observed by Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare33 169–97) that the variational structure of (0.1) differs substantially in the case and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare33 169–97) is restricted to the case which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case with a different variational approach. [ABSTRACT FROM AUTHOR]